A New First Class Algebra, Homological Perturbation and Extension of Pure Spinor Formalism for Superstring

TL;DR

This paper introduces an extended pure spinor formalism for superstring theory by developing a new first class algebra, employing homological perturbation theory, and constructing a nilpotent BRST-like charge with minimal additional ghosts.

Contribution

It presents a novel extension of the pure spinor formalism that removes constraints, using homological perturbation theory to achieve a nilpotent charge with fewer ghosts and systematic vertex operator construction.

Findings

Constructed a nilpotent BRST-like charge $\,\hat{Q}$ with minimal ghosts.

Developed a systematic method for massless vertex operators.

Presented a composite $b$-ghost field satisfying $T(z) = \,\{\,\hat{Q}, B(z)\}$.

Abstract

Based on a novel first class algebra, we develop an extension of the pure spinor (PS) formalism of Berkovits, in which the PS constraints are removed. By using the homological perturbation theory in an essential way, the BRST-like charge $Q$ of the conventional PS formalism is promoted to a bona fide nilpotent charge $\hat{Q}$, the cohomology of which is equivalent to the constrained cohomology of $Q$. This construction requires only a minimum number (five) of additional fermionic ghost-antighost pairs and the vertex operators for the massless modes of open string are obtained in a systematic way. Furthermore, we present a simple composite "$b$-ghost" field $B(z)$ which realizes the important relation $T(z) = \{\hat{Q}, B(z)\} $, with $T(z)$ the Virasoro operator, and apply it to facilitate the construction of the integrated vertex. The present formalism utilizes U(5) parametrization and the manifest Lorentz covariance is yet to be achieved.